# relation between asymptotes and big oh notation

• Jan 14th 2012, 05:56 PM
Jskid
relation between asymptotes and big oh notation
What does big oh notation have to do with asymptotes?
• Jan 14th 2012, 08:50 PM
CaptainBlack
Re: relation between asymptotes and big oh notation
Quote:

Originally Posted by Jskid
What does big oh notation have to do with asymptotes?

If g(x) is an asymptote to f(x), what can we say using big-O notation about f and g?

CB
• Jan 14th 2012, 09:37 PM
Jskid
Re: relation between asymptotes and big oh notation
• Jan 14th 2012, 10:48 PM
CaptainBlack
Re: relation between asymptotes and big oh notation
Quote:
Exactly, but it does not work the other way around.

CB
• Jan 15th 2012, 10:37 AM
emakarov
Re: relation between asymptotes and big oh notation
Quote:
Quote:

Originally Posted by CaptainBlack
Exactly, but it does not work the other way around.

It does not work the other way in the sense that f(x) = O(g(x)) does not imply that g(x) is an asymptote to f(x). However, if g(x) ≠ 0 is an asymptote to f(x), then we have both f(x) = O(g(x)) and g(x) = O(f(x)).
• Jan 15th 2012, 12:00 PM
Jskid
Re: relation between asymptotes and big oh notation
Quote:
Where I'm confused is $x^2$ is $O(x^3)$ but the functions diverge and there's no asymptote?
• Jan 15th 2012, 12:02 PM
emakarov
Re: relation between asymptotes and big oh notation
Quote:

Originally Posted by emakarov
f(x) = O(g(x)) does not imply that g(x) is an asymptote to f(x)

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• Jan 15th 2012, 12:11 PM
Jskid
Re: relation between asymptotes and big oh notation
Quote:

Originally Posted by emakarov
It does not work the other way in the sense that f(x) = O(g(x)) does not imply that g(x) is an asymptote to f(x).

Is it possible that f(x) is an asymptote to g(x) but g(x) is not an asymptote to f(x). If yes I still don't see how $x^3$ is an asymptote to $x^2$ or $x^2$ is an asymptote to $x^3$
• Jan 15th 2012, 12:18 PM
emakarov
Re: relation between asymptotes and big oh notation
According to Wikipedia, asymptote is a straight line. Therefore, neither $x^2$ nor $x^3$ can be an asymptote. Yes, it is possible that f(x) is an asymptote to g(x) but g(x) is not an asymptote to f(x) because (the graph of) f(x) is a straight line and g(x) is not. But the fact that f(x) = O(g(x)) does not imply that either f(x) or g(x) is an asymptote to anything or that that either f(x) or g(x) has an asymptote.
• Jan 15th 2012, 12:27 PM
Jskid
Re: relation between asymptotes and big oh notation
Quote:

Originally Posted by emakarov
According to Wikipedia, asymptote is a straight line. Therefore, neither $x^2$ nor $x^3$ can be an asymptote. Yes, it is possible that f(x) is an asymptote to g(x) but g(x) is not an asymptote to f(x) because (the graph of) f(x) is a straight line and g(x) is not. But the fact that f(x) = O(g(x)) does not imply that either f(x) or g(x) is an asymptote to anything or that that either f(x) or g(x) has an asymptote.

Sorry but why is f(x) a straight line?
• Jan 15th 2012, 12:43 PM
emakarov
Re: relation between asymptotes and big oh notation
An asymptote is a straight line by definition.

Edit: It may happen that f(x) is an asymptote to g(x) for some f(x) and g(x). This implies that f(x) is a straight line. This does not imply that g(x) is a straight line; therefore, this does not imply that g(x) is an asymptote to anything.
• Jan 15th 2012, 12:46 PM
Jskid
Re: relation between asymptotes and big oh notation
Quote:

Originally Posted by emakarov
An asymptote is a straight line by definition.

Oh I think I'm getting it. In f(x) = O(g(x)) f(x) is not the family of functions that are greater than or equal to O(g(x)), it's the upper bound to it?
• Jan 15th 2012, 12:54 PM
emakarov
Re: relation between asymptotes and big oh notation
Quote:

Originally Posted by Jskid
In f(x) = O(g(x)) f(x) is not the family of functions that are greater than or equal to O(g(x))?

Well, f(x) is not a family of functions; it's one function. You may consider O(g(x)) as a family, but not f(x). Second, why would you think that f(x) = O(g(x)) is somehow related to f(x) greater than or equal to g(x)? If anything, the first, very rough, idea behind f(x) = O(g(x)) is that f(x) <= g(x). (In reality, f(x) <= C * g(x) for some constant C, and this inequality has to hold only eventually.)
• Jan 15th 2012, 12:55 PM
Jskid
Re: relation between asymptotes and big oh notation
Here f(n) is O(g(n))
http://img440.imageshack.us/img440/1410/helpbv.png

but I see no asymptote?
• Jan 15th 2012, 01:01 PM
emakarov
Re: relation between asymptotes and big oh notation
Quote:

Originally Posted by emakarov
But the fact that f(x) = O(g(x)) does not imply that either f(x) or g(x) is an asymptote to anything or that that either f(x) or g(x) has an asymptote.

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